finite_of_isArtinianRing — Mathlib

English

For a finite type algebra A over R, A is finite over R iff A is Artinian.

Русский

Для конечного типа алгебры A над R A конечно по R тогда и только тогда Artnian.

LaTeX

$$Module.finite_iff_isArtinianRing$$

Lean4

/-- If `A` is a finite type algebra over `R`, then `A` is an Artinian ring and `R` is Jacobson
implies `A` is finite over `R`. -/
/- If made an instance, causes timeouts synthesizing `FaithfulSMul R I.ResidueField` at
`Ideal.algebraMap_residueField_eq_zero` and `Ideal.ker_algebraMap_residueField` during
simpNF linting. -/

theorem finite_of_isArtinianRing [IsJacobsonRing R] [IsArtinianRing A] : Module.Finite R A :=
  have := finite_of_isSemisimpleRing R (A ⧸ Ring.jacobson A)
  IsSemiprimaryRing.finite_of_isArtinian R A A

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